Related Work.
The best attempts of upper bounds are [1]}, [2]}, [3]}.
In particular, [1]} achieves a regret bound of \(O(d\sqrt{T}\log T)\)  with high probability; [2]} achieves \(O(d\sqrt{T\log T}poly(\log \log T))\)  in expectation and \(O(d\sqrt{T}\log T poly(\log \log T))\)  with high probability; and [3]} achieves \(O(d\sqrt{T\log T}poly(\log \log (T)))\)  in expectation and only when the number of actions is finite and bounded by \(2^{d/2}\).
